Content of ACT Mathematics Test

The ACT Mathematics test questions are drawn from several areas. These content topics represent areas of mathematics commonly taught by the end of grade 11 that are important to success in entry-level college mathematics courses.

Pre-algebra
Elementary algebra
Intermediate algebra
Coordinate geometry
Plane geometry
Trigonometry
Statistics and Probability

1. Pre-Algebra

Pre-algebra questions involve operations such as using whole numbers, fractions, decimals, and integers; numbers raised to positive integer powers and square roots of numbers; ratio, proportion, and percent; multiples and factors of integers; absolute value; ordering numbers from least to greatest or greatest to least; simple linear equations with one variable; simple probability and counting the number of ways something can happen; representing and interpreting data in charts, tables, and graphs; and simple descriptive statistics including mean, median, and mode.

Number and Quantity

Math questions in this category test your knowledge of numbers and fundamental math concepts and operations, including the following:

Perform calculations on whole numbers and decimals
Recognize equivalent fractions and fractions in lowest terms
Locate rational numbers (whole numbers, fractions, decimals, and mixed numbers) on the number line
Recognize single-digit factors of a number
Identify a digit’s place value
Demonstrate knowledge of elementary number concepts, including rounding, ordering of decimals, pattern identification, primes, and greatest common factor
Write powers of 10 using exponents
Comprehend the concept of length on the number line, and find the distance between two points
Understand absolute value in terms of distance
Find the distance between two points with the same x-coordinate or y-coordinate in the coordinate plane
Add, subtract, and multiply matrices (tables of numbers)
Order fractions
Find and use the least common multiple
Demonstrate knowledge of complex numbers and multiply two complex numbers
Comprehend the concept of irrational numbers, such as π
Apply properties of rational exponents
Use relations involving addition, subtraction, and scalar multiplication of vectors and matrices
Analyze and draw conclusions based on number concepts

2. Elementary Algebra

Elementary algebra questions evaluate understanding and skills such as using variables to express relationships, substituting the value of a variable in an expression, performing basic operations on polynomials, factoring polynomials, solving simple quadratic equations (the kind that can be solved by factoring), solving linear inequalities with one variable, and applying properties of integer exponents and square roots.

Algebra knowledge and skills tested include the following:

Demonstrate knowledge of basic expressions, such as b + g to identify a total
Solve equations in the form x + a = b, where a and b are whole numbers or decimals
Use substitution to evaluate mathematical expressions
Combine like terms, such as 2x + 5x
Add and subtract algebraic expressions
Multiply two binomials
Match inequalities with their graphs on the number line
Demonstrate knowledge of slope
Solve real-world problems by using first-degree equations
Solve inequalities
Match linear or compound inequalities with their graphs on the number line
Add, subtract, and multiply polynomials
Solve quadratic equations
Factor quadratics
Work with squares/square roots and cubes/cube roots of numbers
Work with scientific notation
Solve problems involving positive integer exponents
Determine the slope of a line from an equation
Solve linear inequalities when the method involves reversing the inequality sign
Solve systems of two linear equations
Solve absolute value equations and inequalities
Match quadratic inequalities with their graphs on the number line

3. Intermediate Algebra

Intermediate algebra questions ask you to apply your knowledge, skills, and reasoning ability to solve problems that involve more advanced topics of algebra such as the quadratic formula, radical and rational expressions, inequalities and absolute value equations, sequences, systems of equations, quadratic inequalities, functions, matrices, roots of polynomials, and complex numbers.

Questions that involve functions test your ability to do the following:

Understand the concept of a function having a well-defined output value at each valid input value
Extend a given pattern by a few terms for patterns that have a constant increase or decrease between terms or that have a constant factor between terms
Evaluate linear, quadratic, and polynomial functions expressed in function notation at the integer level
Interpret statements that use function notation in terms of their context
Find the domain of polynomial functions and rational functions
Find the range of polynomial functions
Find where a rational function’s graph has a vertical asymptote
Use function notation for simple functions of two variables
Relate a graph to a situation described qualitatively in terms of faster change or slower change
Build functions for relations that are inversely proportional or exponential
Find a recursive expression for the general term in a sequence described recursively
Evaluate composite functions of integer values
Compare actual values and the values of a modeling function to judge model fit and compare models
Demonstrate knowledge of geometric sequences
Demonstrate knowledge of unit circle trigonometry
Match graphs of basic trigonometric functions with their equations
Use trigonometric concepts and basic identities to solve problems
Demonstrate knowledge of logarithms
Write an expression for the composite of two simple functions

Questions that involve both algebra and functions test your ability to do the following:

Solve problems using whole numbers and decimals in the context of money
Solve one- or two-step arithmetic problems using positive rational numbers, such as percent
Relate a graph to a situation described quantitatively
Solve two- or three-step arithmetic problems involving concepts such as rate and proportion, sales tax, percentage off, and estimation
Perform word-to-symbol translations
Solve multistep arithmetic problems that involve planning or converting units of measure (for example, feet per second to miles per hour)
Build functions and write expressions, equations, or inequalities with a single variable for common pre-algebra settings, such as rate and distance problems and problems that involve proportions
Match linear equations with their graphs in the coordinate plane
Solve word problems containing several rates, proportions, or percentages
Build functions and write expressions, equations, and inequalities for common algebra settings
Interpret and use information from graphs in the coordinate plane
Solve complex math problems involving percent of increase or decrease or requiring integration of several concepts
Build functions and write expressions, equations, and inequalities when the process requires planning and/or strategic manipulation
Analyze and draw conclusions based on properties of algebra and/or functions
Analyze and draw conclusions based on information from graphs in the coordinate plane
Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax² + c
Given an equation or function, find an equation or function whose graph is a translation by specified amounts up or down

4. Coordinate Geometry

Coordinate geometry questions deal with the real number line and the standard (x,y) coordinate plane. They involve number line graphs as well as graphs of points, lines, polynomials, circles, and other curves in the standard (x,y) coordinate plane. They also involve relationships between equations and graphs; slope, parallel and perpendicular lines; distance; midpoints; transformations; and conics.

5. Plane Geometry

Plane geometry questions test your grasp of topics that are usually part of high school geometry, though some topics are introduced in earlier course work. Included are the properties and relations of plane figures (triangles, rectangles, parallelograms, trapezoids, and circles); angles, parallel lines, and perpendicular lines; translations, rotations, and reflections; simple three-dimensional geometry; and measurement concepts such as perimeter, area, and volume. Justification, proof techniques, and logical conclusions are a part of this area.

Geometry questions are based primarily on the mathematical properties and relationships of points, lines, angles, two-dimensional shapes, and three-dimensional objects. Knowledge and skills tested include the following:

Estimate the length of a line segment based on other lengths in a geometric figure
Calculate the length of a line segment based on the lengths of other line segments that go in the same direction (for example, overlapping line segments and parallel sides of polygons with only right angles)
Perform common conversions of money and of length, weight, mass, and time within a measurement system (for example, inches to feet and hours to minutes)
Compute the area and perimeter of triangles, rectangles, and other polygons
Use properties of parallel lines to find the measure of an angle
Exhibit knowledge of basic angle properties and special sums of angle measures (for example, 90°, 180°, and 360°)
Use geometric formulas when all necessary information is given
Locate points in the coordinate plane
Translate points up, down, left, and right in the coordinate plane
Use several angle properties to find an unknown angle measure
Count the number of lines of symmetry of a geometric figure
Use symmetry of isosceles triangles to find unknown side lengths or angle measures
Recognize that real-world measurements are typically imprecise and that an appropriate level of precision is related to the measuring device and procedure
Compute the perimeter of composite geometric figures with unknown side lengths
Compute the area and circumference of circles
Given the length of two sides of a right triangle, find the length of the third side
Express the sine, cosine, and tangent of an angle in a right triangle as a ratio of given side lengths
Determine the slope of a line from points or a graph
Find the midpoint of a line segment
Find the coordinates of a point rotated 180° around a given center point
Use relationships involving area, perimeter, and volume of geometric figures to compute another measure (for example, surface area for a cube of a given volume and simple geometric probability)
Use the Pythagorean theorem
Apply properties of 30°–60°–90°, 45°–45°–90°, similar, and congruent triangles
Apply basic trigonometric ratios to solve right-triangle problems
Use the distance formula
Use properties of parallel and perpendicular lines to determine an equation of a line or coordinates of a point
Find the coordinates of a point reflected across a vertical or horizontal line or across y = x
Find the coordinates of a point rotated 90° across a vertical
Recognize special characteristics of parabolas and circles (for example, the vertex of a parabola and the center or radius of a circle)
Use relationships among angles, arcs, and distances in a circle
Compute the area of composite geometric figures when planning and/or visualization is required
Use scale factors to determine the magnitude of a size change
Analyze and draw conclusions based on a set of conditions
Solve multistep geometry problems that involve integrating concepts, planning, and/or visualization

6. Trigonometry

Trigonometry questions cover topics such as the trigonometric ratios defined for right triangles; the values, properties, and graphs of the trigonometric functions; trigonometric identities; trigonometric equations; and modeling with trigonometric functions.

7. Statistics and Probability

Statistics is a branch of mathematics that involves the collection and analysis of large quantities of numerical data. Probability is a branch of mathematics that involves calculating the likelihood of an event occurring or a condition existing. Statistics and probability questions test your ability to do the following:

Calculate averages
Read and extract relevant data from a basic table or chart and use the data in a computation
Use the relationship between the probability of an event and the probability of its complement
Calculate the missing data value given the average and all other data values
Translate from one representation of data to another (for example, from a bar graph to a circle graph)
Compute probabilities
Describe events as combinations of other events (for example, using and, or, and not)
Demonstrate knowledge of and apply counting techniques
Calculate the average given the frequency counts of all the data values
Manipulate data from tables and charts
Use Venn diagrams in counting
Recognize that when data summaries are reported in the real world, results are often rounded and must be interpreted as having appropriate precision
Recognize that when a statistical model is used, model values typically differ from actual values
Calculate or use a weighted average
Interpret and use information from tables and charts, including two-way frequency tables
Recognize the concepts of conditional and joint probability and of independence expressed in real-world contexts
Distinguish among mean, median, and mode for a list of numbers
Analyze and draw conclusions based on information from tables and charts, including two-way frequency tables
Understand the role of randomization in surveys, experiments, and observational studies
Demonstrate knowledge of conditional and joint probability
Recognize that part of the power of statistical modeling comes from looking at regularity in the differences between actual values and model values